Exercise 17.3.8. Given a set A and a partition P = {A₁, A2, A3,…} of A. Let ~p be the equivalence relation associated with the partition P. Now define a function f : A → N as follows: f(a)= 1 2 if a € A₁ if a € A₂ ⠀ n if a € An In general: f(a) = jiff a € Aj. (a) Show that a ~p b ⇒ f(a) = f(b).
Exercise 17.3.8. Given a set A and a partition P = {A₁, A2, A3,…} of A. Let ~p be the equivalence relation associated with the partition P. Now define a function f : A → N as follows: f(a)= 1 2 if a € A₁ if a € A₂ ⠀ n if a € An In general: f(a) = jiff a € Aj. (a) Show that a ~p b ⇒ f(a) = f(b).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 17.3.8 part A and please show step by step and explain
Transcribed Image Text:Exercise 17.3.8. Given a set A and a partition P = {A1, A2, A3,...} of A.
Let ~p be the equivalence relation associated with the partition P. Now
define a function f: A → N as follows:
f(a) =
1
2
⠀⠀
n if a € An
if a € A₁
if a € A₂
In general: f(a) = jiff a € Aj.
(a) Show that a ~pb f(a) = f(b).
(b) Let ~f be the equivalence relation defined from f as in Proposition 17.3.6.
Show that ~p~ by showing that a ~p ba~g b.
Exercise 17.3.8 amounts to a proof of the following proposition.
Proposition 17.3.9. Given a set A with partition P = {A1, A2, A3,...}.
Let ~p be the associated equivalence relation. Then there exists a function
f: A→N with associated equivalence relation such that ~p=~f.
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